Analysis of the controllability from the exterior of strong damping nonlocal wave equations^*

¹ University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, College of Natural Sciences, 17 University AVE. STE 1701, San Juan, PR 00925-2537, USA.
² Universidad de Santiago de Chile, Facultad de Ciencia, Departamento de Matemática y Ciencia de la Computación, Casilla 307-Correo 2, Santiago, Chile.

^** Corresponding author: mahamadi.warma1@upr.edu

Received: 16 October 2018
Accepted: 24 April 2019

Abstract

We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation associated with the fractional Laplace operator subject to the non-homogeneous Dirichlet type exterior condition. In the first part, we show that if 0 < s < 1, Ω ⊂ ℝ^N (N ≥ 1) is a bounded Lipschitz domain and the parameter δ > 0, then there is no control function g such that the following system

is exact or null controllable at time T > 0. In the second part, we prove that for every δ ≥ 0 and 0 < s < 1, the system is indeed approximately controllable for any T > 0 and g ∈ D(O × (0,T)), where O ⊂ ℝ^N \ Ω is any non-empty open set.